Diffusion in the general theory of relativity

TL;DR

This paper extends Markovian diffusion theory into the framework of general relativity, deriving a relativistic Kramers equation and analyzing diffusion processes in curved spacetime and expanding universes.

Contribution

It introduces a new approach using moving orthonormal frames to formulate relativistic stochastic calculus and derives the general relativistic Kramers equation in phase space.

Findings

Diffusion equations transform consistently under hypersurface-preserving coordinate changes.

In the quasi-steady state, the diffusion results align with thermodynamic kinetic theory.

The framework applies to diffusion in expanding universe models.

Abstract

The Markovian diffusion theory in the phase space is generalized within the framework of the general theory of relativity. The introduction of moving orthonormal frame vectors both for the position as well the velocity space enables to bypass difficulties in the general relativistic stochastic calculus. The general relativistic Kramers equation in the phase space is derived both in the parametrization of phase space proper time and the coordinate time. The transformation of the obtained diffusion equation under hypersurface-preserving coordinate transformations is analyzed and diffusion in the expanding universe is studied. It is shown that the validity of the fluctuation-dissipation theorem ensures that in the quasi-steady state regime the result of the derived diffusion equation is consistent with the kinetic theory in thermodynamic equilibrium.